[math-fun] Primogeniture question

Let f(X) be a nonconstant polynomial in Z[X] not of the form f(X) = g(X) h(X) where g(X), h(X) are in Z[X] - {1,-1}. Does there necessarily exist an integer N such that f(N) is a (positive or negative) prime number ? If so, is there known to be a minimum number m(d) of such N, where d = deg(f) ? (What if f(X) is further assumed to be monic?) --Dan